Yesterday someone asked a question in SE about indeterminate quadratic equations(of the form $x^2−ny^2=1$ which got me really interested in them and I thought I would try to learn something related to it. One of the interesting topics that I found was Pell's equation which led me to the Chakravala method.
I was wondering what are the possible applications of these type of quadratics in real life?
If your real life involves higher mathematics, then there are plenty of applications of Pell's equation. For one thing, it's where you get units in real quadratic fields. Also, it's where you get good rational approximations to $\sqrt n$.
A lot of puzzle questions come down to Pell equations as well, things like finding right-angle triangles with integer sides with the two legs differing by 1.
I bet if you do a search for Pell or Pellian at this website you'll find dozens of questions where the answer involved solving a Pellian.